Topic
Segment addition postulate
About: Segment addition postulate is a research topic. Over the lifetime, 4 publications have been published within this topic receiving 28 citations. The topic is also known as: Chasles' relation.
Papers
TL;DR: In this paper, the authors consider a straight line on a flat surface running from point A to C and passing though B, and infer that the shortest distance between A and C is ten inches.
Abstract: [FIRST PARAGRAPH]
Consider a straight line on a flat surface running from point A to C and passing though B. Suppose the distance AB to be four inches, and the distance BC to be six inches. We can infer that the distance AC is ten inches. Of all geometrical inferences, this is surely one of the simplest. Of course, things are a little more complicated if the surface is not flat. If A, B and C are points on a sphere, then the shortest distance between A and C may be smaller (it may even be zero). We can make our inference immune from concerns about non-Euclidean spaces, however, by qualifying it as follows: if AB = n, and BC = m, then, in the direction A⇒B⇒C, the distance AC is n + m. This is apparently entirely trivial. But trivial truths can hide significant ontological ones. Let us translate our mathematical example to the physical world, and suppose A, B and C to be points, still in a straight line, but now at the centre of gravity of three physical objects:
14 citations
1 Jan 1991
TL;DR: This chapter discusses formula for calculating distance from a point to a line, which gives efficient formulae for the distance d 1 between a point P and the line defined by points A and B , and thedistance d 2 between P andThe line segment AB .
Abstract: This chapter discusses formula for calculating distance from a point to a line. It gives efficient formulae for the distance d 1 between a point P and the line defined by points A and B , and the distance d 2 between P and the line segment AB . An example of this application is an interactive program searching for a displayed line segment nearest to a cursor being moved by the user. For this type of operation, coordinates usually are integers, and computationally expensive functions need to be avoided to provide rapid response. Often, finding d 1 2 or d 2 2 is sufficient. The distance from a point to a line is the length of the segment PQ perpendicular to line AB . When the perpendicular intersection point Q is outside the segment AB , d 1 is shorter than d 2 —the distance from P to the nearest point of the segment. For applications only focusing on segments within some maximum distance from P , a simple bounding box test can be used to quickly reject segments that are too far away.
9 citations
1 Jan 2010
TL;DR: In this paper, the Steiner-Lehmus theorem holds for all points P on the line AJ of a triangle ABC with AB 6 AC, and for points on the extension of AJ that are not very far away from side BC.
Abstract: For any point P in the plane of the triangle ABC, we let BBP, CCP be the cevians through P. Then the Steiner-Lehmus theorem states that if I is the incenter of ABC and if BBI = CCI then AB = AC. Letting the internal angle bisector of A meet BC at J, it is stated in (13) that the same holds if I is replaced by any point on the ray AJ. However, the proof there is valid for points on segment AJ and for points on the extension of AJ that are not very far away from side BC. In this paper, we consider all points P on the line AJ and we answer the question whether BBP = CCP implies AB = AC, or equivalently whether AB 6 AC implies BBP 6 CCP. For a triangle ABC with AB 6 AC, we describe a line segment XY on the line AJ inside of which there exists P with
7 citations
TL;DR: This paper presents an O (n log n) algorithm for finding a central link segment of P that generalizes previous results for finding an edge or a segments of P from which P is visible and in the same time bound finds a central links segment of minimum length.
Abstract: A central link segment of a simple n-vertex polygon P is a segment s inside P that minimizes the quantity maxx∈Pminy∈sdL(x,y), where dL(x,y) is the link distance between points x and y of P. In this paper we present an O (n log n) algorithm for finding a central link segment of P. This generalizes previous results for finding an edge or a segment of P from which P is visible. Moreover, in the same time bound, our algorithm finds a central link segment of minimum length. Constructing a central link segment has applications to the problems of finding an optimal robot placement in a simply connected polygonal region and determining the minimum value k for which a given polygon is k-visible from some segment.
2 citations
Performance Metrics
| Year | Papers |
|---|---|
| 2010 | 1 |
| 2004 | 1 |
| 2000 | 1 |
| 1991 | 1 |